Think of some examples satisfying the premises of these propositions. Do all of them satisfy the conclusion? If not, then you have a counterexample. If so, do you know why?
are these statements true or false? give a reason or a counterexample to justify your answers.
(a) A square invertible matrix A with complex number entries is diagonalizable just when inverse of A is diagonalizable.
(b)Every diagonalizable matrix whose entries are complex numbers is normal.
(c)Every square matrix has a kernel dimension 0.
(d)There is a 3x3 matrix A with real number entries so that the kernel of A is 2-dimensional and the kernel of A-I is also 2-dimensional.
(e)A square invertible matrix A with real number entries is diagonalizable just when inverse of A is diagonalizable.
(f)If A is a normal matrix with complex number entries and A^2=0 then A=0.
(g)Any square matrix with complex number entries is diagonalizable.
(h)Suppose that A is an nxn matrix with real number entries. Suppose that Ax has the same ength as x, for every vector x in R^n. Then A is orthogonal.
(i)Suppose that A is symmetric matrix. The eigenvalues of A^2 are the squares of the eigenvalues of A.
(j)Every diagonalizable matrix whose entries are real numbers is symmetric.
(k)If a square matrix A id diagonalizable, then A^2 is diagonalizable.
(l)Every invertible matrix has kernel of dimension 0.